Topology optimization of supporting structures for seismic response reduction of spatial structures

Transcription

1 Proceeg of the ISS ual Sympoum 06 Spatal Structure the t Cetury 6 0 September, 06, Tokyo, Japa K. Kawaguch, M. Ohak, T. Takeuch (e.) Topology optmzato of upportg tructure for emc repoe reucto of patal tructure uj MIZ*, Makoto OHSKI a, Seta TSD b * Departmet of rchtecture, Hrohma verty Hgah-Hrohma , Japa myazu@hrohma-u.ac.jp a Kyoto verty b Okayama Prefectural verty btract flexble upportg tructure reucg emc repoe of two type of patal tructure, a arch a a lattce ome, are propoe. The upportg tructure of the arch a the lattce ome are moele a two- a three-meoal tru tructure, repectvely, a ther topology a croectoal area are optmze olvg olear programmg problem. The optmzato problem of the arch ve to two mall optmzato problem to obta the optmal oluto effcetly. It emotrate throu tme-htory repoe aaly that the flexble upportg tructure ca reuce the emc repoe of the roof tructure, a tallg vcou amper to the upportg tructure effectve to further reucto of the repoe. Keywor: arch, lattce ome, flexble upportg tructure, topology optmzato, emc repoe. Itroucto Varou emc repoe cotrol ytem, for example, tff emc eg, bae olato, a pave eergy pato ytem have bee propoe to reuce emc amage of bulg tructure. Some of them are effectve alo to patal tructure uch a arche, ome a hell; however, t rather ffcult to cotrol emc repoe of patal tructure becaue everal moe houl be coere the proce of repoe evaluato. The ma emc amage of patal tructure caue by recet earthquake are the fall of otructural compoet uch a celg a hagg equpmet, a the amage at the coecto betwee the roof a the upportg tructure. Oe of the metho to cotrol thee amage to reuce the accelerato repoe a the erta force of the roof tructure. The prevou tuy by Ohak et al. [] how that ug the flexblty of a bae tructure effectve reucg roof placemet a accelerato of a bulg frame. Myazu et al. [] how that the ormal repoe of the roof of a arch ca be reuce by flexble upportg tructure obtae throu topology optmzato metho. I th tuy, we apply the cocept of the flexble tructure to the upportg tructure of two type of patal tructure: a arch a a lattce ome, a we optmze the topology a the hape of the upportg tructure. The arch a the lattce ome are moele a two- a three-meoal tructure, repectvely, a the topology, the hape, a the cro-ectoal area of the upportg tructure moele a pjote true are optmze ug a olear programmg approach. The accelerato repoe the ormal recto a the erta force of the roof tructure evaluate by the complete quaratc combato (CQC) metho are mmze uer cotrat tffe uer elf-wet. It how umercal tuy that the flexble upport reucg roof repoe are uccefully geerate by the Copyrt 06 by uj MIZ, Makoto OHSKI, Seta TSD Publhe by the Iteratoal ocato for Shell a Spatal Structure (ISS) wth permo.

2 Proceeg of the ISS ual Sympoum 06 Spatal Structure the t Cetury propoe optmzato metho. Effectvee of tallg vcou amper to the flexble upportg tructure alo emotrate throu tme-htory repoe aalye.. Flexble upportg tructure for arch. Overvew of flexble upport for arch I a arch upporte by covetoal tff tructure, a how Fg. (a), t kow that the ormal rectoal repoe excte the roof tructure eve whe the arch ubjecte oly to horzotal emc grou moto. Sce the ormal repoe of the roof caue the bucklg of roof member a fallg of o-tructural member [], reucg the ormal repoe of the roof tructure oe of the mportat pot the emc eg of arche. Fgure (b) how the cocept of the reucto of the ormal rectoal repoe by utlzg the flexblty of the upportg tructure. By ug the upportg tructure whoe top oe move maly the tagetal recto of the roof, a how by thck arrow Fg. (b), t expecte that the ormal rectoal repoe of the roof reuce. I our prevou tuy [], the flexble upportg tructure for a arch geerate throu two tep of tatc a yamc topology optmzato. I th tuy, we optmze ot oly the topology but alo the hape of the upportg tructure the yamc optmzato problem to obta better oluto. I Secto., the reult of the tatc optmzato are ummarze for completee of the paper. Normal recto Tagetal recto Supportg tructure Roof tructure Semc put Semc put (a) Fgure : (a) arch wth tff upportg tructure; (b) a arch wth flexble upportg tructure.. Statc optmzato problem The tatc optmzato problem for a upportg tructure olve to obta the upportg tructure whoe top oe move the agoal recto uer lateral loa a how Fg (a). The covetoal grou tructure approach ue for topology optmzato. The grou tructure ha 0 oe clug upport a 9 tru member whch are ot coecte at ther terecto wthout oe. The ma of 4000 kg attache at oe 0 to repreet the ma of a roof tructure. The wet of the tru member gore. (b) Drecto of placemet 0 ateral loa P m m m m (a) Fgure : (a) Grou tructure for tatc topology optmzato; (b) topology of the optmal oluto; (c) fal topology of the optmal oluto. (b) (c)

3 Proceeg of the ISS ual Sympoum 06 Spatal Structure the t Cetury et hv a hh eote the placemet the - a - recto, repectvely, of oe 0 uer a lateral loa P = 7.84 kn, whch correpo to 0 % of wet of the ma at oe 0. The objectve fucto to be maxmze formulate a hv R () hh Deg varable are the cro-ectoal area of all m (= 9) tru member, whch are eote by a 6 vector,, ). The lower a the upper bou of are.0 0 a ( m.0 0 (m ), repectvely. et a eote the - a - rectoal placemet, repectvely, uer elf-wet. I orer that the upportg tructure ha eou vertcal tffe a oe t have too mall lateral tffe, the lower bou of 0. 0, , a 0.06 (m) are e. The optmzato problem calle Problem formulate a follow: hh Problem: maxmze ubject to R( ) hh ( ) ( ) ( ), (,, m) fter olvg Problem, we olve the problem calle Problem to remove ueceary member. Problem, whch mmze the total volume V() of tru member uer placemet cotrat, formulate a follow: Problem : mmze ubject to V ( ) ( ) ( ) ( ) hh R( ) CR hh, (,, m) where R opt eote the optmal oluto of Problem. Note that R opt multple by a coeffcet C (= 0.95) to obta a lower bou to e uffcet large feable rego. Optmzato problem a are olve ug the optmzato lbrary SNOPT Ver. 7 [4], whch ue equetal quaratc programmg. fte fferece approach ue to calculate the etvty coeffcet. The frame aaly oftware OpeSee [5] ue for tructural aaly. Fgure (b) how the bet optmal oluto wth R() = 0.57 amog te optmal oluto obtae from raomly geerate te fferet tal oluto. The cro-ectoal area of the member cate by ahe le have ther lower bou. The wth of the ol le proportoal to t cro-ectoal area. By removg the member wth lower bou cro-ectoal area a oe 4, 5, a 6, whch are utable oe, we obta fal topology wth R() = how Fg. (c). Note that the th member coectg oe a 7 ca be maufacture a a prg.. Dyamc optmzato problem I th ecto, we optmze the cro-ectoal area of the th member a the hape of the upportg tructure by olvg yamc optmzato problem coerg teracto betwee the roof a the upportg tructure. Fgure (a) how a arch wth covetoal tff upportg tructure. The roof tructure cot of te teel beam member wth oug moulu hh opt () ()

4 Proceeg of the ISS ual Sympoum 06 Spatal Structure the t Cetury N/mm. The cro-ectoal area a eco momet of area of the beam are m a m 4, repectvely. The cro-ectoal area of the tru member of the upportg tructure.0 0 m. Noe 0 a 0 are coecte by te bar whoe cro-ectoal area.0 0 m. The ma of 800 kg a 400 kg are attache at oe 0 to 0 a at the oe the upportg tructure except for upport, repectvely; thu, the total ma of th arch 600 kg. We call th arch tffmoel. Fgure (b) how the moe hape of the t moe of the tff-moel. The atural pero of the t a the moe are 0.69 a 0.94, repectvely. It ee that the eformato of the roof much larger tha that of the upportg tructure m 9.5 m (a) Fgure : (a) arch wth tff upportg tructure (tff-moel); (b) moe hape of the t moe. Fgure 4(a) how a arch upporte by the flexble tructure obtae Secto.. The upportg tructure are locate ymmetrcally to make the roof move the maly tagetal recto a llutrate Fg. (b) uer horzotal emc exctato. The ma of 800 kg attache at oe the upportg tructure exclug t upport o that th arch ha the ame total ma a the tffmoel. (b) Fgure 4: (a) arch wth flexble upportg tructure; (b) hape varable of the upportg tructure; (c) moe hape of the t moe. Sce t ha bee fou throu prelmary tme-htory aaly that the cro-ectoal area of member cate by ahe le Fg. 4(a) play a mportat role o the reucto of accelerato repoe of the roof tructure, the cro-ectoal area of member choe a a eg varable the followg optmzato problem. The varable of the upportg tructure, whch efe the hape of the upportg tructure to mofy the movg recto of oe 0 a 0, llutrate Fg. 4(b). The objectve fucto to be mmze evaluate by where F N 9 9 (4), u, N Sa ( T, h ) r r Sa ( Tr, hr ) r u (5) r 4

5 Proceeg of the ISS ual Sympoum 06 Spatal Structure the t Cetury r r hr ) 8 hhr h 4hhr ( hr h ) (6) ( 4h )( 4h ( ) 4h h ( ) 4( h h ) r r T for T Sa( T, h).4 for 0.6 T h.074 / T for T I Eq. (4)-(7), u the abolute accelerato repoe the ormal recto of the th oe evaluate by the CQC metho [6], r the moal-correlato coeffcet betwee the th a the rth moe, a, h, T, a are the partcpato factor, the ampg factor, the atural pero, a the ormal rectoal compoet of the th moe, repectvely. The rato of the atural crcular frequece of the rth moe to that of the th moe eote by χ. Equato (7) efe the eg accelerato repoe pectrum for a mle level earthquake. I the CQC metho, the umber of moe N 4, a the moal ampg efe a Rayle ampg wth h h et a eote - a - rectoal placemet of the top oe of the left upportg tructure uer elf-wet. I orer that the upportg tructure ha eou vertcal tffe, lower bou 0. 0 a (m) are age. Note that the lower bou relaxe to 0 % of that of Problem to have eou large feable rego. The lower a upper bou a for are the ame a thoe for Secto., a thoe for are 0. 5 m a 0.5 m, repectvely. Hece, the optmzato problem formulate a: Problem : mmze ubject to F(, ) (, ) (, ) The cro-ectoal area a the hape varable are optmze to f the optmal value m a 0. 5 m, repectvely. The optmal objectve value.5 m/, whch 46 % of that of the tff-moel. Th oluto calle flexble-moel. The moe hape of the t moe of the flexble-moel, llutrate Fg. 4(c), how that the large eformato prouce the upportg tructure. The atural pero of the t a the moe are.06 a 0.5, whch are.9 a.8 tme a large a thoe of the tff-moel..4 Tme-htory repoe I th ecto, tme-htory aalye o the tff- a the flexble- moel are coucte ug the oftware OpeSee to obta the maxmum repoe uer grou moto. The eg accelerato pectrum efe by Eq. (7) how a thck le Fg. 5(a). Te emc grou moto are geerate to be compatble wth the eg repoe pectrum ug raom phae. The accelerato repoe pectra of te grou moto are how th gray le Fg. 5(a). The tme cremet a the urato of each grou moto are 0.0 a 60, repectvely. The tme htory of oe of the grou accelerato how Fg. 5(b). Fgure 6(a) a (b) how the mea value of the maxmum abolute accelerato the tagetal a the ormal recto of the oe. The oe are etfe by -coorate. The maxmum ormal accelerato repoe amog all the oe of the flexble-moel 44 % of that of the tff- (7) (8) 5

6 Proceeg of the ISS ual Sympoum 06 Spatal Structure the t Cetury moel; however, ce the upportg tructure optmze to move the tagetal recto, the maxmum tagetal accelerato of the flexble-moel creae to 48 % of that of the tff-moel. ccelerato (m/ ) Pero () (a) ccelerato (m/ ) Tme () (b) Fgure 5: (a) ccelerato repoe pectra (h=0.05). Thck le: eg pectrum; th le: pectra of te grou moto.; (b) tme htory of a grou accelerato. m/ Tagetal recto Normal recto m m N/m Fgure 6: (a) Tagetal accelerato repoe; (b) ormal accelerato repoe ( : Flexble-moel wth amper, : Flexble-moel wthout amper, : Stff-moel); (c) relato betwee c a accelerato repoe..5 Itallato of vcou amper I orer to reuce the tagetal accelerato creae by the flexble upportg tructure, we tall vcou amper betwee the par of oe 7, 8, a 7, 8 the upportg tructure how Fg. 4(a), whch have large relatve placemet uer emc exctato. Fgure 6(c) how the relato betwee the ampg coeffcet c of the amper a the accelerato repoe uer the grou moto how Fg. 5(b). The orate the orm of accelerato repoe efe Eq. (4). It ee that the amper wth c N/m ecreae the tagetal accelerato repoe wthout creag the ormal accelerato repoe. The plot of upermpoe o Fg. 6(a) a (b) are the reult of the flexble-moel wth the amper whch have c = 0000 N/m. It cofrme that the vcou amper uccefully ecreae the tagetal accelerato wthout creae of the ormal accelerato.. Flexble upportg tructure for lattce ome m/. Overvew of flexble upport for lattce ome The lattce ome coere th tuy, how Fg. 7, ha a roof tructure wth tralatoal urface, whch upporte by four upportg tructure at the corer of the roof. I recet earthquake, falure at the coecto betwee the roof a the upportg tructure oe of the ma tructural amage of ome; therefore, t aume that reucto of erta force of the roof tructure effectve to avo the amage at the coecto ue to earthquake exctato. I the cae of ome, ce the behavor uer emc grou moto complex becaue of three meoal repoe, t ffcult to prect the effectve eformato charactertc of the upportg tructure to reuce the erta force of the roof. I th tuy, therefore, we try to optmze 6 m/

7 Proceeg of the ISS ual Sympoum 06 Spatal Structure the t Cetury the upportg tructure throu oly yamc optmzato problem about roof+upportg tructure. Vcou amper are talle to the optmze upportg tructure to further reuce the erta force of the roof tructure. Note that the repoe uer oe rectoal emc grou moto coere th tuy. Supportg tructure Roof tructure Semc put Fgure 7: attce ome upporte by four upportg tructure. Formulato of optmzato problem The tructural moel of the lattce ome wth tff upportg tructure how Fg. 8(a) a (b), where (,, Z) a (u, v, w) cate the global coorate a the local coorate of the roof member, repectvely. The roof tructure cot of arche the - a -recto, repectvely, a thee arche are coecte a rg jot at the terecto. The pa of the arch 9.5 m a the het of the top of the ome from the top of upportg tructure 5. m. The cro-ectoal area a the eco momet of area wth repect to v- a w-ax of the roof member except for the ege arche are m, m 4, a m 4, repectvely, a thoe of the member of the ege arche are m, m 4, a m 4, repectvely. Each upportg tructure cot of 0 p-jote tru member whoe cro-ectoal area are m. The ma of 000 kg a 600 kg are attache at the top oe a the other oe, repectvely, of the upportg tructure, a the ma of 800 kg attache at the oe of the roof tructure. We call th lattce ome tff-moel. I th ecto, we optmze the topology a the cro-ectoal area of the upportg tructure to reuce the erta force of the roof tructure uer -rectoal emc moto. Therefore, the objectve fucto to be mmze evaluate by F I N N mk Sa( T, h ) rmk rr Sa( Tr, hr ) (9) r k k where m k a eote the ma of oe k a the -rectoal compoet of the th moe, repectvely. The value of 7, whch the umber of oe o the roof tructure except for the oe of the corer. The umber of moe N 50 th problem. The ampg factor h h are ue for Rayle ampg. Fgure 8(c) how the moe hape of the t moe of the tff-moel. It ee that the eformato of the roof tructure omat the moel. The atural pero of the t a the 4th moe are a 0., repectvely. The objectve fucto FI N. 7

8 m m Proceeg of the ISS ual Sympoum 06 Spatal Structure the t Cetury ege arch Z 9.5 m ege arch 9.5 m w u Z v m m Fgure 8: (a) Stff-moel; (b) upportg tructure; (c) moe hape of the t moe. The wth of the le proportoal to the cro-ectoal area of the member. et gx, gy, a gz eote the -, -, a Z- rectoal placemet of the top oe of the upportg tructure, repectvely, how by re flle crcle Fg. 8(a) a (b). The lower bou gx 0.0, gy 0. 0, a gz (m) are e to each placemet to avo too mall tffe agat elf-wet. The cro-ectoal area of m (=) tru member how Fg. 9(a) a the varato of locato of the oe of the upportg tructure how by re arrow Fg. 9(b) are choe a eg varable, whch are eote by vector (,, m ) a calar. The 5 lower a the upper bou of (,, m) are.0 0 m a.0 0 m, repectvely, a thoe of are 0. 5 m a 0. 5 m, repectvely. Note that the topology of the upportg tructure ymmetry wth regar to the axe cate by re ahe le Fg. 9 (a) a (c). We formulate the optmzato problem calle Problem 4 a follow. Problem 4 : mmze ubject to F (, ) I gx gy gz (, ) (, ) (, ) gx gy gz, (,, m) (0) (0) () () () (5) (4) 4 () 5 () (7) (6) () (9) 7 (8) 6 Fgure 9: (a) Noe a member umber of the upportg tructure; (b) varato of locato of the oe of the upportg tructure; (c) top vew of the moel.. Optmzato reult Fgure 0(a) a (b) how the topology of the optmal oluto obtae by olvg Problem 4 whch ha the tal oluto wth (,, m) a = 0. The wth of the le proportoal to t 8

9 Proceeg of the ISS ual Sympoum 06 Spatal Structure the t Cetury cro-ectoal area, a the member that reach cro-ectoal area of ther lower bou are cate by ahe le. The cro-ectoal area of all the member how by ol le reach ther upper bou. The value of ha t lower bou. The objectve fucto N, whch about 78 % of that of the tff-moel. Fgure 0(c) how the moe hape of the moe whch ha omat eformato the -recto. The atural pero.9, whch 96 % of the t moe atural pero of the tff-moel. Note that the t moe ha the rotatoal eformato wth repect to the Z-ax a the atural pero of.. It ee that the -plae eformato of the roof maller a that of the upportg tructure larger compare wth thoe of the tff-moel. Z Fgure 0: (a) Topology of the optmal oluto; (b) upportg tructure; (c) the moe hape of the moe. Sce the upportg tructure how Fg. 0(b) ha ueceary member a oe, we remove member 5, 6, 8 a oe 4, a replace member a coecte to oe 4 wth oe member. By olvg Problem 4 wth m =, the cro-ectoal area (,) of member a how Fg. (a) a are optmze to f optmal value m a -0.5 m, repectvely. The objectve fucto N, whch 6 % of that of the tff-moel. We call th moel flexble-moel. The moe hape of the moe wth atural pero.5, how Fg. (b), almot the ame a Fg. 0(c). Fgure (c) how the relato betwee the objectve fucto F I a the cro-ectoal area of member a, whch cate that mall cro-ectoal area of member a are better for reucto of F I a log a the cotrat are atfe. Z 5 () () 7 6 Z Fgure : (a) Topology of flexble-moel; (b) moe hape of the moe; (c) relato betwee the objectve fucto a the cro-ectoal area of member a..4 Repoe reucto by vcou amper ee Fg. (b), the relatve placemet betwee oe a 7 large the upportg tructure; therefore, tallg a vcou amper betwee the two oe aume to be effectve o patg a part of emc put eergy. Fgure (a) how the relato betwee the ampg coeffcet c of the vcou amper wth F I whch the mea value of the maxmum erta force of the roof tructure the -recto, obtae by tme htory repoe aaly ug te grou 5 moto. It cofrme that F I ha the mallet value arou c.50 N/m. Fgure (b) how the tme htory repoe of the erta force of the roof tructure of the tff-moel a the flexble- 9 F I (N).5E+05.0E+05.5E+05.0E E+04 optmal oluto } ot atfy the cotrat 0.0E E+00.0E E E E-05.0E-04.E-04 cro-ectoal area of member a (m )

10 Proceeg of the ISS ual Sympoum 06 Spatal Structure the t Cetury 5 moel wth vcou amper of c.50 N/m. The amper uccefully ecreae the erta force repoe all over the urato tme of the grou moto. F I (N).0E+05.8E+05.6E+05.4E+05.E+05.0E E E E+04.0E E E+00.00E+06.00E+06.00E+06 c (N/m) (a) -.0E tme () Fgure : (a) Relato betwee c a F I ; (b) tme htory of the erta force of the roof tructure. Blue le: tff-moel, re le: flexble-moel wth vcou amper. 4. Cocluo. The flexble upportg tructure of the arch, whch make the roof tructure move maly the tagetal recto to reuce the ormal accelerato repoe of the roof, uccefully geerate throu two tep of tatc a yamc optmzato problem.. I the lattce ome, the upportg tructure whch reuce the erta force of the roof tructure uer oe rectoal emc grou moto obtae by olvg yamc optmzato problem.. Itallg vcou amper to the flexble upportg tructure very effectve to further reuce the emc repoe of the roof tructure of the arch a the lattce ome. ckowlegemet Th reearch upporte by JSPS KKENHI Grat Number The prelmary umercal aaly Secto by Mr. Tetuto Taguch, former grauate tuet of Hrohma verty, alo apprecate. Referece [] Ohak M, Iwatuk O, Wataabe H. Semc repoe of bulg frame wth flexble bae optmze for revere rockg repoe. Egeerg Structure, 04; 74: [] Myazu, Ohak M, Tua S. Topology optmzato of upportg tructure for emc repoe reucto of a arch. SCIENCE CHIN Techologcal Scece, 06; 59: [] Jot Etoral Commttee for the Report o the Great Eat Japa Earthquake Dater. Report o the Great Eat Japa Earthquake, Dater Bulg Sere Volume ( Japaee). Tokyo: Maruze Publhg Co., [4] Gll P E, Murray W, Sauer M. SNOPT: a SQP algorthm for large-cale cotrae optmzato. SIM J Opt, 00, : [5] Ope Sytem for Earthquake Egeerg Smulato (OpeSee). Pacfc Earthquake Egeerg Reearch Ceter, verty of Calfora, Berkeley [6] Wlo E, Der Kurea, Bayo E P. replacemet for the SRSS metho emc aaly. Earthequake Eg Struct Dy, 98, 9: Ierta force of the roof tructure the -recto (N).0E+05.0E+05.0E E E E+05 (b) 0